Research on Estimating Unknown Functions in a Class of Integral Inequalities Involving Discontinuous Functions

Integral inequalities serve as fundamental analytical tools for investigating the qualitative behavior of solutions to differential, integral, and integro-differential equations. Classical inequalities such as those of Gronwall, Bellman, and Bihari have been widely used to establish uniqueness, boundedness, and stability results. However, their applicability is limited in the presence of discontinuities, impulsive effects, and nonlinear terms of power type. To address these limitations, this paper develops a new class of integral inequalities specifically designed for discontinuous functions. The proposed framework unifies delayed arguments, impulsive jumps, and power-function nonlinearities within a single inequality structure. By combining interval partitioning, piecewise analysis, and mathematical induction, together with extensions of the Gronwall–Bellman–Bihari approach, we derive explicit maximal bounds for unknown functions that satisfy these inequalities. These bounds generalize many existing results as special cases, while maintaining concise and computationally convenient forms. The explicit estimates enhance the theoretical tractability of discontinuous systems and offer practical applicability in complex settings. To demonstrate the effectiveness of the proposed results, we apply them to impulsive differential equations with nonlinear growth and delayed feedback. The example shows that the new inequalities guarantee boundedness even under strong nonlinear effects and abrupt impulses, thus confirming the robustness of the framework. The results highlight potential applications in control theory, stability analysis, and dynamical systems involving memory, delays, or impulsive disturbances. The contributions of this study are threefold: (i) it establishes a unified inequality framework that simultaneously incorporates delay, impulse, and nonlinear structures; (ii) it provides constructive explicit bounds that improve upon existing results; and (iii) it lays a foundation for extending the analysis to fractional-order, stochastic, or multidimensional systems. Overall, this research advances the qualitative theory of differential equations and enriches the available tools for analyzing discontinuous dynamical models.